# What is a non-artificial example of a bifunctor that is contravariant in both arguments?

The Hom functor is a bifunctor that’s covariant in one argument and contravariant in the other.

One can view the construction of group rings as a bifunctor that is covariant in both arguments.

But what is a (hopefully not contrived) example of a bifunctor that is contravariant in both arguments?

• I'm assuming you would consider $(V,W) \mapsto (V \oplus W)^*$ as a bifunctor from $\text{Vect} \times \text{Vect} \to \text{Vect}$ "contrived"? Dec 12, 2020 at 4:54
• If you write it as bilinear maps $V \times W \to k$ I think it's not contrived at all! Dec 12, 2020 at 6:01
• @HallaSurvivor You may as well offer it as a solution, because you never know if your idea will be the most natural one that occurs to anybody. Dec 12, 2020 at 17:16

Given any covariant bifunctor $$A \times B \to C$$ and a contravariant functor $$C \to D^\text{op}$$ we can compose to get a contravariant bifunctor.
One simple example of this might be $$(V, W) \mapsto (V \oplus W)^*$$ which sends a pair of vector spaces to the dual of their direct sum.
If we do the same construction with the tensor product instead of the direct sum, then we can write this in a slicker way by looking at bilinear maps $$V \times W \to k$$, which of course gives you $$(V \otimes W)^*$$. This is a nice observation due to Qiaochu, and is a natural object to study.